This contribution addresses the estimation of invertible processes in a functional time series setting. While asymptotic upper bounds for the estimation error of such operators for specific ARMA processes in the Hilbert space of squareintegrable functions have been considered in recent years, this paper adds to the existing research by designing consistent estimates for the operators defining an invertible representation of a stationary process in a general separable Hilbert space under mild conditions that hold for many classes of functional time series. These results imply in turn consistency results with explicit rates for related operator estimates for Hilbert space-valued causal linear processes, as well as functional MA, AR and ARMA processes.